Download F From Vibrating Strings to a Unified Theory of All Interactions

Barton Zwiebach
From Vibrating
Strings to a Unified
Theory of All Interactions
F
or the last twenty years, physicists have investigated
String Theory rather vigorously. The theory has revealed
an unusual depth. As a result, despite much progress in our understanding of its remarkable properties, basic features of the theory
remain a mystery. This extended period of activity is, in fact, the
second period of activity in string theory. When it was first discovered in the late 1960s, string theory attempted to describe strongly
interacting particles. Along came Quantum Chromodynamics—
a theory of quarks and gluons—and despite their early promise,
strings faded away. This time string theory is a credible candidate
for a theory of all interactions—a unified theory of all forces and
matter. The greatest complication that frustrated the search for such
a unified theorywas the incompatibility between two pillars of twentieth century physics: Einstein’s General Theory of Relativity and
the principles of Quantum Mechanics. String theory appears to be
30 ) zwiebach
mit physics annual 2004
the long-sought
quantum mechanical theory of gravity and
other interactions. It is almost
certain that string theory is a
consistent theory. It is less certain that
it describes our real world. Nevertheless,
intense work has demonstrated that string
theory incorporates many features of the physical
universe. It is reasonable to be very optimistic about
the prospects of string theory.
Perhaps one of the most impressive features of string theory is the appearance of
gravity as one of the fluctuation modes of a closed string. Although it was not discovered exactly in this way, we can describe a logical path that leads to the discovery
of gravity in string theory. One considers a string, similar in manyways to the vibrating strings with tension and mass that are studied in freshman physics. This time,
however, the string is relativistic. This means that the classical mechanics of this
string is consistent with Einstein’s special theory of relativity. A relativistic string
is, in fact, a very interesting and subtle object with a rich spectrum of vibration modes.
These classical vibrations, however, cannot be identified with physical particles.
But quantum theory comes to the rescue: the quantum mechanics of the relativistic
string gives vibration modes that can be identified with physical particles! A
particular quantum vibration mode of the closed string describes a graviton, the
quantum of the gravitational field. A particular quantum vibration of an open string
describes a photon, the quantum of the electromagnetic field. It is the magic of quantization that makes these results possible. In string theory all particles—matter
particles and force carriers — arise as quantum fluctuations of the relativistic
string. Physicists struggled to invent a quantum theory of gravity during much
of the twentieth century, and the answer came from the quantization of classical
mit physics annual 2004 zwiebach
( 31
relativistic strings. We are reminded of an opinion expressed by Dirac in 1966 [1]:
“The only value of the classical theory is to provide us with hints for getting
a quantum theory; the quantum theory is then something that has to stand
in its own right. If we were sufficiently clever to be able to think of a good
quantum theory straight away, we could manage without classical theory
at all. But we’re not that clever, and we have to get all the hints that we can
to help us in setting up a good quantum theory.”
The purpose of this article is to explain some of the unusual features of relativistic
strings and to show one way in which string theory may describe the Standard Model
of particle physics.
What are relativistic strings?
To gain some understanding of relativistic strings, we can compare them with the
more familiar nonrelativistic strings. Nonrelativistic strings are typically characterized by two independent parameters: a string tension T0 and a mass per unit
length m 0. Each of the four strings on a violin, for example, has a different tension
and mass density. When a string with fixed endpoints is also static, the direction
along the string is called the longitudinal direction. Such a string can exhibit small
transverse oscillations (Figure 1a). In this case, the
velocity of any point on the string is orthogonal to
the longitudinal direction. The velocity n of a transverse wave moving along the string is a simple function of the tension and the mass per unit length:
(a)
n== T0 /m 0 .
(b)
Figure 1
(a) In a transverse oscillation the motion
of any point on the string is perpendicular
to the longitudinal direction.(b) In a
longitudinal oscillation the motion of any
point on the string (represented by a thin
slinky) is along the direction of the string.In
order to detect longitudinal motion we must
be able to tag the points along the string.
(1)
A nonrelativistic string may support a different
type of oscillation. When we have a longitudinal
oscillation, the velocity of any point on the string
remains along the string (Figure 1b). In a longitudinal oscillation the wave velocity does not involve the
tension, but rather a tension coefficient that describes
how the tension changes upon small stretching of the string. More important, a longitudinal wave requires the existence of structure along the string. In order to tell that
the various points of the string are really oscillating we must be able to tag them.
If this is not possible, a longitudinal oscillation is undetectable because, as a whole,
the string does not move. Transverse motion is less subtle; we can always tell when
the string moves away from the equilibrium longitudinal direction.
It takes a significant amount of imagination to construct the classical mechanics of relativistic strings. In fact, the mechanics is simplest for the so-called “massless relativistic string.”This is the string that one quantizes to obtain string theory.
To gain intuition, let’s discuss four surprising properties of these strings.
(1) The relativistic string is characterized by its tension T0 alone—there is no
independent mass density parameter. The velocity of transverse waves on
continues on page 46
32 ) zwiebach
mit physics annual 2004
)
String Theory for Undergraduates?
The Story Behind 8.251
When, in the fall of 2001, distinguished string
theorist Professor Barton Zwiebach first proposed to
the Physics Education Committee a new elective for
the Department’s undergraduate curriculum based
upon his upcoming textbook, “A First Course in String
Theory,”the response was a moment of startled
silence. How, the Committee members wondered,
could an area of physics be taught to undergraduates
that was built upon intellectual concepts viewed as
challengingly opaque by not a few of the faculty?
before one could try to understand, even on a basic
level, what string theory is about. [ Thus] it was very
impressive, and intellectually very satisfying, to see
from Zwiebach’s class that basic knowledge about
classical and quantum mechanics is sufficient to get
a head start in this subject....To make this theory
accessible to students at the undergraduate level
can hardly be overestimated in its importance.”
— Martin Zwierlein, Graduate Student, Atomic Physics
“Originally I decided to take the class because
Nevertheless, Zwiebach’s talent and reputation as
one of the most gifted instructors at MIT was wellestablished, so the funds were granted to develop the
new elective, “8.251: String Theory for Undergraduates.”
Launched in the Spring 2002 semester and repeated
annually, the class continues to attract an equal number
of undergraduate and graduate students. It has been
so successful that Zwiebach received the 2002 Everett
Moore Baker Memorial Award for Excellence in
Undergraduate Teaching, the only MIT prize whose
winner is chosen solely by undergraduates.
Comments from his students show a keen appreciation
of both the topic and the teacher.
“The class itself was often fantastic. Midway
through the term, when we’d reached Chapter 10 in
his book, Prof. Zwiebach announced that we had
done three semesters of quantum field theory in one
lecture. It was a heady feeling...”
“I’ll always be grateful for 8.251. Unlike most
classes around here, it left a warm and fuzzy spot in
my heart. It has had a practical payoff, too: learning to
handle commutator relations early gave me a jump
over my 8.05 [Quantum Physics II] compadres, and
seeing Lagrangian dynamics early let me delve into
journal articles with less trepidation. I had a great time
with my 8.06 [Quantum Physics III] term paper, mostly
because Prof. Zwiebach’s class introduced me to
fruitful new concepts I could then turn around and
apply elsewhere, giving me that spine-tingling shiver
of knowledge fitting together.”
— Blake C. Stacey (SB ’04)
“Barton Zwiebach’s course finally bridges the gap
between theoretical physics as taught on the undergraduate level and its current frontier, string theory.
Before taking this course, I was convinced one would
need to learn very sophisticated mathematical tools
string theory is...on the frontier of physics, and this
class proposed to teach me the subject (at least some
parts of it) with a minimum of previously required
knowledge...”
“The class itself was a novel way of teaching
the topic and...quite different from the way string
theory is taught in other texts. Instead of beginning
with abstract field theoretic concepts, 8.251 started
rooted in the physics that we were all familiar with: the
mechanics of a simple string. It all started there and
quickly went through many iterations until arriving at
the quantum mechanics of relativistic strings. Though
at times the math was difficult, as is unavoidable in this
subject, the concepts were very clear throughout the
journey, which also included a discussion of the theory
of branes, T-duality and a few exotic topics like string
thermodynamics and black holes.”
“One thing that is for certain is that the class
would simply not be the same without Prof. Zwiebach;
his clear lecturing and willingness (and ability) to
answer questions was great. I enjoyed the class to the
point of volunteering to help look over the chapters
of the textbook that were yet to be written, because I
really wanted to see more material on this subject.”
“All in all, the class was very exciting; it was
unlike most other physics classes at MIT and remains
among my favorites.”
— Alan M. Dunn (SB ’04)
For a more detailed look at 8.251: STRING THEORY
FOR UNDERGRADUATES , visit the class home page at
http://mit.edu/8.251/www/. The class textbook, A First
Course in String Theory [2], is available from Cambridge
University Press (http://publishing.cambridge.org/
stm/physics/strings/).
mit physics annual 2004 zwiebach
( 33
Barton Zwiebach: From Vibrating Strings to a Unified Theory of All Interactions
continued from page 32
n =c
this string is the velocity of light c, so using (1) the mass
density m 0 is fixed once T0 is fixed:
c == T0 /m
n =c
Figure 2
A relativistic open string can rotate rigidly
about its midpoint.The angular velocity
must be such that the endpoints move
with the speed of light.
0
’ m 0=T0 /c 2.
(2)
Special relativity tells us that mass and energy are
interchangeable, but familiar examples involve quantum
processes, such as massive particles that annihilate into
energetic (zero-mass) photons. In the relativistic string,
energy/mass conversion occurs classically. Imagine beginning with an infinitesimally short relativistic string and
stretching it out to some length L. Since the string tension
is constant, the work done on the string is equal to the
product T0 L of the tension times the length. This energy
makes up the rest mass of the string. Energy is converted
into rest mass by stretching the string! The mass is equal
to the energy divided by c 2, so it equals T0 L/c 2. Consequently, the mass per unit length is T0 /c 2, as anticipated in (2). The relativistic string
has no intrinsic mass; the mass arises from work done against the tension.
(2) The relativistic string does not support longitudinal oscillations. This is a
revealing fact: it tells us that the string has no substructure. The points along
the relativistic string cannot be tagged in an unambiguous way. When a string
moves a little, we cannot really tell which point went where. There is a minor
exception: if we have an open string, we can keep track of the motion of the
endpoints, which, after all, are points. Many times people ask, What is the
string made of? The lack of longitudinal oscillations tells us that no meaningful answer can be provided: the classical relativistic string has no
constituent parts that can be identified.
(3) The endpoints of a free relativistic open string move with the speed of
light. For familiar strings, oscillations require that the motion of the
endpoints be constrained. The simplest constraint is to fix the endpoints; the
string can then have a nonzero tension and oscillations are possible. Nontrivial motion is possible for relativistic open strings even if the endpoints are
not fixed. Elementary mechanics suggests that the effective tension of the
string must vanish at the endpoints. This is actuallyachieved when the endpoints
move at the speed of light. One of the simplest open string motions is that
of an open string that rotates rigidly about its midpoint (Figure 2). This motion
has an unusual property: the angular momentum J of this string is linearly
proportional to the square of the energy E of the string:
J=a' E 2
(3)
The constant of proportionalitya' is called the slope parameter. The above propertywas the reason why physicists attempted (and still attempt!) to use some kind
of string theory to describe strongly interacting particles. Indeed, hadronic
resonances fit rather accurately a linear relation between angular momentum
46 ) zwiebach
mit physics annual 2004
and the square of the mass (or the square of the energy). This relation is completely
unusual: for a rigid bar rotating nonrelativistically about its midpoint, one finds
the rather different J ; =E . Equation (3) can be understood roughly by assuming that the mass of the string is concentrated at the endpoints. Since the speed of
the endpoints is constant and equal to the speed of light, the angular momentum
is proportional to the length of the string times the mass of the string. Given that
both the length and the mass of the string are proportional to its energy, the
angular momentum is proportional to the energy squared.
(4) A relativistic string has an orientation which determines the sign of the string
charge. Consider an electron and its antiparticle, the positron. They are oppositely charged point particles. Being zero-size points, there is no intrinsic
geometrical property that distinguishes their charges. This is different in
string theory. Relativistic strings come with an orientation. For a closed string,
the orientation is an arrow that defines a preferred direction along the
string. One can travel along a closed string in two directions; the orientation picks one out of these two (see Figure 3). It turns out that oppositelyoriented
strings have opposite string charges. In contrast to the case of point particles, in string theory the sign of charge has a geometrical basis. While string
charge is a novel concept, the implications for open strings are readily
understood. To specify an open string we must choose a direction or draw
an arrow along the string. This arrow creates a clear-cut distinction between
the two, previously similar, endpoints: the arrow points away from one
endpoint, called the beginning endpoint, and towards the other endpoint,
called the final endpoint. A surprising effect then takes place: the string charge
forces the open string endpoints to acquire opposite electric charges! String
Figure 3
Relativistic strings carry orientation,a
direction of travel along the string indicated
by arrows. Top line: two oppositely oriented
closed strings are states with opposite string
charge.Bottom line: two oppositely oriented
open strings.The endpoints of open strings
carry ordinary electric charge.The charges at
the open string endpoints are opposite:(+)
at the final endpoint and (–) at the
beginning endpoint.
+
-
-
+
mit physics annual 2004 zwiebach
( 47
charge transmutes into electric charge. The orientation points from the negatively charged to the positively charged endpoint. Since open strings carry
electric charges, we may attempt to identify known charged particles with
excitations of open strings.
The above properties, derived in the classical theory of strings, remain true in
the quantum theory of strings. Further surprises emerge, however, when relativistic strings are quantized. One finds that quantum mechanical strings cannot
propagate consistently in spacetimes of arbitrary dimensionality. For the simplest —
bosonic strings—the spacetime must be twenty-six dimensional. For superstrings,
strings whose excitations include bosons and fermions, the dimensionalityof spacetime is ten, one of time and nine of space. Quantization also implies that strings have
quantum states of oscillation. This allows us to identify the oscillations of strings
with particles, which are themselves quanta of familiar fields. The masses of the
particles associated with string oscillations are computed using the quantum theory.
While closed string oscillations that could be identified with gravitons have positive mass in the classical theory, their mass turns out to be exactly zero in the quantum theory! This is precisely what is needed, since gravitons are exactly massless
particles. There was no reason to expect gravity to arise from fluctuating strings,
but it does. Quantum relativistic strings provide a theoryof quantum gravity. A related
effect occurs for open strings: massless oscillations of open strings represent photons.
Building blocks of the Standard Model
There are four known forces in nature. The Standard Model of particle physics
summarizes the present-day understanding of three of them. It describes the
electromagnetic force, the weak force and the strong force, but leaves out the
gravitational force. The Standard Model also describes the elementary particles
that have been discovered so far.
The electromagnetic force is transmitted by photons, the quanta of the electromagnetic field. The weak force is responsible for the process of nuclear beta decay,
in which a neutron decays into a proton, an electron and an anti-neutrino. The
strong force or color force holds together the constituents of the neutron, the
proton, the pions and many other subnuclear particles. These constituents, called
quarks, are held so tightly by the color force that they cannot be seen in isolation.
In the late 1960s the Weinberg-Salam model of electroweak interactions put together
electromagnetism and the weak force into a consistent, unified framework. The
theory is initially formulated with four massless particles that carry the forces. A
process of symmetry breaking gives mass to three of these particles: the W +, the
W –, and the Z 0. These particles are the carriers of the weak force. The particle that
remains massless is the photon. The theory of the color force is called quantum
chromodynamics (QCD). The carriers of the color force are eight massless particles, colored gluons that, just as the quarks, cannot be observed in isolation. The
quarks respond to the gluons because they carry color; in fact, quarks come in three
colors. The electroweak theory together with QCD form the Standard Model of
particle physics.
48 ) zwiebach
mit physics annual 2004
Since we aim to show how the familiar particles and interactions may arise in
string theory, we now summarize the particle content of the Standard Model. We
have already said that gravity appears automatically in string theory as a fluctuation of closed strings. Therefore, we will not focus on gravity, but rather on the other
force carriers and the matter particles, both of which arise from vibrations of
open strings.
The Standard Model includes twelve force carriers: eight massless gluons,
the W+ , W–, Z 0, and the photon. All of them are bosons. There are also many matter
particles, all of which are fermions. The matter particles are of two types: leptons
and quarks. The leptons include the electron e– , the muon m – , the tau t– , and the
associated neutrinos ne , nm , and nt. We can list them as
Leptons : (ne , e–), (nm , m –), and (nt, t–).
(4)
Since we must include their antiparticles, this adds up to a total of twelve
leptons. The quarks carry color charge electric charge and respond to the weak
force, as well. There are six different types or “flavors”of quarks: up (u), down (d),
charm (c), strange (s), top (t), and bottom (b). We can list them as
Quarks : (u, d), (c, s), and (t, b).
(5)
The u and d quarks, for example, carry different electric charges and respond
differently to the weak force. Each of the six quark flavors listed above comes in
three colors, so this gives 63 3 = 18 particles. Including the antiparticles, we get a
total of 36 quarks. Adding leptons and quarks together we have a grand total of
48 matter particles.
Although the matter particles displayed above and some of the gauge bosons
have masses, these masses are in some sense remarkably small. Consider the
fundamental constants of nature: Newton’s gravitational constant, the speed of light
and Planck’s constant. Since there are three basic units—those of mass, length and
time—there is a unique way to construct a quantity with the units of mass using
only the three fundamental constants. The resulting mass is called the Planck mass
and its numerical value is about 2.23 10 – 5 grams. While ordinary by the standards
of macroscopic objects, this prototype mass is extraordinarily large when compared
with the masses of elementary particles: it is twenty-two orders of magnitude
larger than the mass of the electron, for example. It is in this sense that elementary particles are essentially massless.
The chirality of the electroweak interactions guarantees that the matter particles cannot acquire masses until electroweak symmetry breaking takes place. If
one adjusts the scale of electroweak symmetry breaking to be small, the matter
particles will be light. To understand the meaning of chirality, we recall that
particles with spin are described in terms of left-handed and right-handed states.
If the spin angular momentum points along the direction of the motion, the particle is said to be right-handed; if the spin angular momentum points opposite to
the direction of motion, the particle is said to be left-handed. A left-handed electron, for example, is denoted as eL– and a right-handed electron is denoted as eR– . The
mit physics annual 2004 zwiebach
( 49
Figure 4
red
Top:Three parallel D-branes (shown as
horizontal lines) are needed to produce the
color interactions.The branes can be labeled
by colors:red,blue and green.The lefthanded quarks are open strings that end on
the colored branes.A red quark,for example,
is a string that ends on the red brane.The
left-handed antiquarks are open strings that
begin on the colored branes.Bottom:The
open strings that begin and end on the brane
configuration are gauge bosons.This brane
configuration supports nine gauge bosons,
eight of which are the gluons of QCD.
blue
green
red
quark
blue green
quark quark
anti-quarks
red
blue
green
electroweak interactions are chiral because the left-handed states and the righthanded states of the Standard Model particles respond differently to the weak forces;
there is a fundamental left-right asymmetry. If we focus on the electron and the
neutrino, for example, we have:
neL
(6)
, eR– , neR .
( )
The left-handed states in the doublet feel the weak interactions, while the
right-handed electron and the right-handed neutrino states do not. A similar
situation holds for the quarks. The left-handed states of the u and d quarks feel
the weak interactions while the right-handed states do not:
uL
(7)
dL , uR , dR .
( )
The existence of mass requires couplings between left- and right-handed states
that are not allowed as long as chirality holds.
D-branes and the Standard Model
D-branes are extended objects in string theory. Whenever we have open strings
we also have D-branes, since the endpoints of open strings must lie on them.
D-branes come in various dimensionalities. A Dp-brane is a D-brane that has p
spatial dimensions. A D2-brane, for example, may look like a sheet of paper, and
a D1-brane may look like a string. In four-dimensional spacetime, a D3-brane may
fill the full extent of the three spatial dimensions, in which case we have a spacefilling brane. Since D-branes extend in various dimensions we can imagine
observers that live on D-branes.
50 ) zwiebach
mit physics annual 2004
String endpoints carry electric charge so, in order to represent a charged particle, we arrange to have a string with one endpoint lying on the D-brane. The other
string endpoint must lie on another, possibly separate D-brane, otherwise the
string would represent two oppositely charged particles, for a net of zero charge.
A positively charged particle is represented by a string that ends on the D-brane,
while a negatively charged particle is represented by a string that begins on the
D-brane. In fact, the photons that couple to these charges arise from open strings
with both endpoints on the D-brane. As required, these states have zero charge.
How can we get quarks using D-branes and strings? Since color is just a
whimsical label for a kind of charge, to obtain three types of color we simply use
three different D-branes: a red brane, a blue brane, and a green brane (Figure 4).
To represent quarks we use strings that have one endpoint on a colored brane while
the other endpoint lies on a different collection of branes, to be specified later. A
string that ends on a green brane is a green quark, a string that ends on a blue brane
is a blue quark, and a string that ends on a red brane is a red quark. Strings that
begin on the colored branes are antiquarks. On the other hand, the gluons—the
carriers of the color force—arise from strings that begin and end on the colored
branes. With three D-branes, there are a total of nine such strings. Out of these,
eight of them are the gluons we are looking for.
For any quark, one endpoint of the corresponding string lies on a color brane.
Where does the other endpoint lie? The answer becomes clear once we consider
the weak interactions. In order to produce the four gauge bosons of the electroweak interactions we need two new D-branes, two “weak branes.”To draw the
brane configuration, it is convenient to use a plane and represent the D-branes as
lines. We take the weak branes to intersect the color branes, as shown in Figure 5.
Let’s now consider the two flavors of quarks indicated in (7). Since the left-handed
u and d quarks feel the weak interactions (in addition to the color force)
the strings that represent them must
have their other endpoint on a weak
brane. Given that we have two weak
branes, we have a perfect fit: the lefthanded u quarks are strings stretched
from one of the weak branes to the
color branes, while the left-handed d
red, blue and green
down quarks
quarks are strings stretched from the
other weak brane to the color branes.
The strings that represent the
left-handed quarks stretch from one
kind of brane to the other. The string
tension forces the strings to have
the minimum possible length, which
weak
branes
in this case is zero, so they represent
massless states that live at the points
Figure 5
The color branes intersect the two weak
branes (shown vertically).Open strings
localized near the intersection that stretch
from the weak branes to the color branes
are left-handed quarks.The two flavors of
quarks (u and d) arise because there are
two weak branes.
red
blue
green
red, blue and green
up quarks
mit physics annual 2004 zwiebach
( 51
Figure 6
The full D-brane configuration in which open
strings represent the familiar particles of the
Standard Model.The vertical branes to the
right are called “right branes” because they
support the right-handed quarks (which do
not feel the weak interactions).The horizontal
branes at the bottom are called “leptonic
branes” since they support the left-handed
leptons (to the left) and the right-handed
leptons (to the right).
gluon
color
branes
dL
UL
UR , d R
W+, W –
eR, nR
nL
leptonic
branes
eL–
weak
branes
right
branes
where the branes intersect. Chirality is a property that guarantees that mass
cannot be readily acquired. When D-branes intersect, there is no small displacement of the branes that eliminates the intersection, so the strings that stretch
from one brane to the other cannot acquire mass. Chirality is a property of states
that arise at brane intersections.
How do we get the right-handed quarks listed in (7)? Since these states feel
the color force, the strings have one end on the color branes. Since they do not feel
the weak interactions, they cannot have their other endpoint on the weak branes
and consequently we need new branes. These new “right branes” must intersect
the color branes for the states to be massless. As shown in Figure 6, the right-handed
quarks stretch from the right branes to the color branes.
Let’s now consider the lepton doublet that includes the left-handed neutrino
and electron [see (6)]. These particles feel the weak interactions, so they are represented by strings that have one endpoint on the weak branes. Since they do not feel
the strong interactions, the other endpoint in those strings must end on new
“leptonic branes.” The left-handed leptons are shown as strings localized at the
intersection of the weak branes and the leptonic branes. Finally, let’s consider the
right-handed electron eR– and the right-handed neutrino neR . These particles feel
neither the color force nor the weak force. They are represented by strings that
stretch from right branes to leptonic branes, as shown at the bottom right corner
of Figure 6.
We have exhibited the states that comprise one family of the Standard
Model. The Standard Model has two additional families, with states completely
analogous to those described in (6) and (7). These are obtained with additional
intersections. The first models to give the precise spectrum of the Standard Model
were constructed in 2001 by Ibanez, Cremades and Marchesano [3].
52 ) zwiebach
mit physics annual 2004
We have so far imagined the D-branes as D1-branes that are stretched on a twodimensional plane. Let us finally show how the brane configuration fits into a tendimensional superstring theory. A physical setup requires an effective four-dimensional
spacetime, so six of the spatial dimensions must curl up into a compact space of
small volume. To visualize the brane configuration we assume that two out of the
six extra dimensions are curled up into a two-dimensional torus (Figure 7). The
D-branes are all chosen to be D4-branes, and three out of their four spatial directions fill our space. The last direction is chosen to appear as a line on the two-dimensional compact torus. So, in fact, Figure 6 was a picture of the D-branes as seen on
the torus, a close-up that does not quite show how the D-branes are fullywrapped
around the torus. The strings that represent the Standard Model particles are localized at the brane intersections and are perceived as particles.
While there are string constructions that give precisely the matter content of
the Standard Model, no one claims to have a derivation of particle physics from
strings. For this, one must also show that symmetry breaking works out correctly
and particles acquire their familiar masses. This has not yet been done. I hope,
however, to have demonstrated that familiar features of our observed universe can
emerge from string theory.
Outlook
We maywonder what are the possible outcomes of an exhaustive search for a realistic string model. One possible outcome (the worst one) is that no string model
reproduces the Standard Model. This would rule out string theory. Another possible outcome (the best one) is that one string model reproduces the Standard
Model. Moreover, the model represents a well-isolated point in the space of all string
solutions. The parameters of the Standard Model are thus predicted. The number
of string models may be so large that a strange possibility emerges: there may exist
many string models with almost identical properties, all of which are consistent with the Standard Model to the
accuracy that it is presently known. In
this possibility there may be a significant loss of predictive power. Other
outcomes may be possible.
New experimental input will also
help us determine if string theory
describes our universe. The recent
discoveryof a nonzero positive cosmological constant has suggested new
directions of investigation based on
quark
cosmological properties of strings. A
discovery of supersymmetry would be
a strong indication that string theory is
Figure 7
The intersecting D-brane configuration.
To visualize a compactification we must
imagine that a compact torus,such as the
one shown in the figure,exists on top of each
point of ordinary three space (represented as
a plane with coordinates X1,X2,and X3).The
D4-branes fill three-space and have one
direction along the torus.The D-branes
appear as lines on the torus.A left-handed
quark is an open string that stretches from
weak to color branes on the torus.It is
perceived as a particle in three-space.
x4 –x5
x 1, x 2, x 3
mit physics annual 2004 zwiebach
( 53
correct because supersymmetry is generic in string theory—it is almost a prediction. The discoveryof extra dimensions, perhaps surprisingly large ones, would also
have dramatic implications. Most likely, finding out if string theory describes our
universe will require a greater masteryof the theory. String theory is in fact an unfinished theory. Much has been learned, but there is no complete formulation of the
theory and its conceptual foundation remains largely mysterious. String theory is
an exciting research area because the central ideas remain to be found.
references
[1] Lectures in Quantum Field Theory, P.A.M. Dirac. Monograph Series, Belfer Graduate
School of Science, Yeshiva University. New York, 1966. See Section 8, entitled “Value
of the Classical Theory.”
[2] A First Course in String Theory, B. Zwiebach. Cambridge University Press, UK, 2004.
[3] Cremades, D., Ibanez, L.E. and Marchesano, F. “More about the Standard Model at
Intersecting Branes.” arXiv:hep-ph/0212048.
barton zwiebach is a Professor of Physics at the Massachusetts Institute of Technology.
He was born in Lima, Peru, where he earned an undergraduate degree in Electrical
Engineering from the Universidad Nacional de Ingenieria in 1977.
Zwiebach’s graduate work was in Physics, at the California Institute of Technology. He
received his Ph.D. in 1983, working under the supervision of Murray Gell-Mann. Zwiebach
has held postdoctoral positions at the University of California, Berkeley, and at MIT, where he
became an Assistant Professor of Physics in 1987, and a tenured member of the faculty in 1994.
54 ) zwiebach
mit physics annual 2004